A non-Hopfian ascending HNN-extension of a finitely presented Hopfian group

2025/12/09 by Kim, Jan, Kim, Junseok, Lee, Yoonjin
#20E06 #20E26 #20F67 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2512.09135

Abstract

We find a non-Hopfian ascending HNN-extension of a finitely presented Hopfian group by providing an explicit construction. This result addresses an analogous question to the one posed by Sapir and Wise, which asks whether there is a non-residually finite ascending HNN-extension of a finitely presented residually finite group. Such an analogy is motivated by Mal'cev's result that every finitely generated residually finite group is Hopfian.

Citations

• A recipe for constructing non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups
• Peripheral fillings of relatively hyperbolic groups
• Elementary subgroups of relatively hyperbolic groups and bounded generation

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• A criterion for HNN extensions of finite p-groups to be residually p
• Embedding Partial HNN Extensions In Ascending HNN Extensions
• Non-linear, solvable, residually p groups
• A non-Hopfian relatively hyperbolic group with respect to a Hopfian subgroup